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Relationship and transfer between mental andwritten arithmeticCatherine Thevenot a & Caroline Castel aa University of Geneva, FAPSE, Geneva, SwitzerlandPublished online: 25 Nov 2011.

To cite this article: Catherine Thevenot & Caroline Castel (2012): Relationship and transfer between mental andwritten arithmetic, Journal of Cognitive Psychology, 24:3, 286-294

To link to this article: http://dx.doi.org/10.1080/20445911.2011.617302

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Relationship and transfer between mental and writtenarithmetic

Catherine Thevenot and Caroline Castel

University of Geneva, FAPSE, Geneva, Switzerland

In a first experiment, adults practiced single- and two-digit mental addition over a 6-day period. Therewas a clear training effect for both types of problems, even if two-digit additions were different from oneday to another. Moreover, participants were tested on their written calculation abilities before and afterthe training programme. We showed that participants who entered the mental arithmetic trainingprogramme did not progress more in written arithmetic than participants who did not receive anytraining between the pre- and the post-tests. Conversely, in a second experiment, participants weretrained in multidigit written addition and we examined the effect of such training on single- and two-digitmental addition. Again and trivially, there was a clear effect of training on written addition, but, moreimportantly, a transfer on mental addition. The implications of these results on the nature of therelationship between mental and written arithmetic are discussed.

Keywords: Addition; Arithmetic abilities; Numerical cognition; Practice.

Given suitable instructions and practice, adultsand children have the ability to acquire almostany skill. Influential models in the literature havemainly focused on two aspects of skill acquisition,namely improvement in speed and reduction oferrors (Anderson, 1982, 1987; Logan, 1988). Inthe domain of mental arithmetic, it is now wellestablished that it is possible to train mentalarithmetic abilities through repetitive practice(Delazer et al., 2003, for example). Both brainimaging and behavioural studies lead to theconclusion that performance improvement aftertraining results both from higher efficiency ofresult retrieval and speed-up and optimisation ofcalculation processes (Imbo & Vandierendonck,2008; Pauli et al., 1994).

However, an even more important issue foreducation is related to transfer of training. As

noted recently by Taatgen, Huss, Dickison, andAnderson (2008), expertise, which can be con-sidered as the end product of skill acquisition, isassociated not only with speed and accuracy butalso with flexibility. Flexibility is the ability toapply a skill to new problems that are differentfrom the problems that served as the basis fortraining. Therefore, in the domain of mentalarithmetic, the question of transfer within andbetween operations is a crucial one. In fact,successful transfer is the key for efficient arith-metic. If the effects of training are limited to theset of operations trained, then practice in arith-metic is not an optimum method of learning. Anoptimum method should enable individuals to usetheir acquired knowledge in a flexible way, whichwould allow new problem solving. Whereasseveral studies have focused on transfer between

Correspondence should be addressed to Catherine Thevenot, FAPSE, University of Geneva, 40, bd du Pont D’Arve, CH-1205

Geneva, Switzerland. E-mail: catherine.thevenot@unige.ch

This work was supported by an FNRS Project Grant (ref. 100014-126919) and by the Ernst & Lucie Schmidheiny Foundation.

We would like to thank Patrick Bruderlein, Sonja Doswald, Isabelle Gay-Dembinski, and Ombeline Goy for collecting and

analysing the data. Finally, we would like to thank Patrick Lemaire, Mark Ashcraft, and an anonymous reviewer for their thoughtful

suggestions on a first version of this manuscript.

JOURNAL OF COGNITIVE PSYCHOLOGY, 2012, 24 (3), 286�294

# 2012 Psychology Press, an imprint of the Taylor & Francis Group, an Informa businesshttp://www.psypress.com/ecp http://dx.doi.org/10.1080/20445911.2011.617302

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problems (from multiplication to division, e.g.,Ischebeck, Zamarian, Schocke, & Delazer, 2009,or addition to subtraction, e.g., Campbell &Agnew, 2009), surprisingly no study has addressedthe question of transfer within a specific opera-tion (from one-digit to two-digit problems, forexample). In fact, in most studies the effect oftraining is assessed by comparing the solutiontimes of the specific set of trained problems to aset of untrained problems. If the trained set issolved better or faster that the untrained set, theauthors can conclude that the training was effi-cient. However, while this approach is informa-tive about the effect of training within the setunder study, it cannot give any information abouttransfer. Without a baseline relative to solvingperformance on untrained problems, it is notpossible to assess whether or not these problemsare solved better after than before training.

Therefore, the first aim of this study is todetermine if practising a specific set of additionstransfers to other sets of additions. For thispurpose, our participants were presented dailywith the same set of single-digit addition pro-blems and were also presented with sets of two-digit additions that differed from one day toanother. In this way, and contrary to previousstudies, it was possible to evaluate the progressmade by participants on the untrained sets ofproblems or, in other words, evaluate the transfereffect within addition problems.

The second aim of this study is to determine ifpractising mental addition transfers to writtenaddition and, conversely, if practising writtenaddition transfers to mental addition. Even if itis common sense to assume that mental andwritten calculation share common components,neuropsychological evidence shows that writtencalculation can be dissociated from other numer-ical abilities. Indeed, Semenza, Miceli, and Girelli(1997) have reported the case of a patient whosedifficulties were limited to multidigit calculationto be solved in written form throughout theapplication of multistep algorithms. On the con-trary, the patient’s abilities in arithmetic factretrieval or complex mental arithmetic werewell preserved. As suggested by other works(for example, Girelli & Delazer, 1996; Temple,1991), it is inferable from Semenza et al.’s resultsthat specific difficulties with written multidigitcalculation stem from a partially distorted knowl-edge of the required procedures. Of course, asalready stated above, this is not to say that somearithmetical components are not common to

multidigit written calculation and mental arith-metic. Indeed, good performance in both domainsnecessarily relies on quick and efficient arithmeticfact retrieval. Moreover, the knowledge of calcu-lation procedures is required to solve written andmental multidigit calculation. However, theseprocedures are not necessarily the same. Inmental multidigit addition, two main strategieshave been identified in adults and children(Lemaire & Arnaud, 2008; Lemaire & Callies,2009; Lucangeli, Tressoldi, Bendotti, Bonanomi,& Siegel, 2003). First, the full decompositionstrategy consists in decomposing both tens andunits (e.g., 53�44�50�40�90; 3�4�7;90�7�97). Second, the partial decompositionstrategy consists in decomposing only the secondaddend (e.g., 34�65�34�60�94; 94�5�99).Lemaire and Callies (2009) showed that bothchildren and adults use the partial decompositionstrategy more often when problems are presentedhorizontally than vertically. Moreover, childrenand adults use the two strategies equally oftenwhen the problem is presented vertically. Inwritten addition, the common strategy is the onetaught at school consisting in adding the units,then the tens, and so on, using carrying whennecessary (Lucangeli et al., 2003). It is thereforeworth investigating whether or not transferbetween mental and written calculation is possi-ble through the practice of shared componentsbetween the two abilities or, on the contrary, if atransfer is impossible because of written calcula-tion specificities. Such approach is based onMorris, Bransford, and Franks’ (1977) ‘‘transferappropriate processing’’ theory, according towhich the value of a particular type of acquisitionmust be defined relatively to the type of activitiesto be performed at the time of the test. Therefore,the existence of common cognitive processesunderlying written and mental arithmetic shouldlead to transfer of learning. On the contrary, if theknowledge acquired during training is not re-quired or involved during testing, then no transferof learning should be observed.

In order to answer these questions, we carriedout two different experiments. In the first one,adults were trained on mental single and multi-digit additions over a 6-day period. Whereassingle-digit additions were the same from oneday to another, different multidigit additions werepresented every day. As in previous studies, welogically predicted an effect of training on single-digit problems. Using different multidigit pro-blems allowed us to determine whether or not

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training could generalise to new problems. More-over, our participants were tested using the well-known written arithmetic fluency test (i.e., subtestof the French Kit; French, Ekstrom, & Price,1963) before and after the training period. It wasthen possible to determine if practising mentalarithmetic transfers to written calculation. Inorder to ensure that potential score improvementon the French Kit was not only due to a test�retest effect, the results obtained by participantstrained on mental arithmetic were compared tothe results obtained by a control group ofparticipants who were not trained between thetwo testing sessions. Conversely, in a secondexperiment, participants were trained on writtenmultidigit additions similar to those used in theFrench Kit over a 6-day period and we examineda potential transfer of such training on mentaladditions.

EXPERIMENT 1

Method

Participants

One hundred and forty-six students from theUniversity of Geneva were paid for their partici-pation. They were ranked as a function of theirarithmetic score on the addition subtest of theFrench Kit. One-third of the students whoobtained the highest scores was considered ashigh-skilled participants and was excluded fromthe experiment. This group was composed of 47participants and their mean score on the additionsubset was 36.47 (SD�5.36; range�31�54). Theother participants were considered as lowerskilled individuals and entered the experimentaldesign. We chose to select only lower-skilledstudents in order to maximise the chance toobserve performance improvement in solvingthe problems. Without this effect, it would havebeen impossible to study possible transfers. Thisgroup was composed of 99 participants and theirmean score was 22.60 (SD�5.26; range�10�30).The 52 participants who decided to continue theexperiment were assigned to one of the group(arithmetic training or control). The arithmeticgroup was composed of 26 participants with amean score of 23.77 and the no-training groupwas composed of 26 participants with a meanscore of 24.46 on the addition subtest of theFrench Kit.

Material and procedure

The written arithmetic test: Subtest of the FrenchKit. Participants completed the addition subtestof the French Kit, both at the pre- and post-testphases. It consists of two pages of 60 additions incolumns involving three numbers of either one ortwo digits (e.g., 63�99�5). All participants aregiven 2 minutes per page and are instructed tosolve the problems as fast and accurately aspossible. The additions correctly solved on eachpage are summed to yield a total arithmetic score.In order to base our analysis on the mean solutiontimes taken by participants to solve one writtenaddition, this total number was divided by 2minutes.

The mental arithmetic training programme.Participants in the arithmetic group were trainedover a 6-day period and the last training dayalways took place before the post-test. Every day,participants had to solve the two same blocks ofone-digit addition problems. These blocks werealways the same from one day to the other.Moreover, they were presented daily with twodifferent blocks of two-digit additions, whichdiffered systematically from day to day. Thepresentation of one- and two-digit blocks wasalternated, always starting with a one-digit block.One-digit blocks comprised the 56 problemscorresponding to the 100 basic additions exclud-ing the additions involving 0 and 1 and tieproblems. Two-digit blocks comprised 28 pro-blems. Half of them required a carry and theother half did not. Moreover, in half of theproblems, the higher operand was placed firstand, in the other half, second. Again, additionsinvolving 0, 1, and ties were discarded from thematerial. Therefore, participants solved 168 addi-tions per day.

Each trial began with the presentation of afixation cross. After the problem was presentedon screen, participants had to enter their answeron the numerical keyboard. If an incorrect answerwas given, a message appeared on screen and thesame addition was presented again until thecorrect response was provided. Solution timescorresponded to the time elapsed between thepresentation of the problem and the validation ofa correct answer by the participant. Feedbackconcerning the total solution time for a block wasgiven at the end of each block. The programmewas controlled by the E-Prime software and eachtraining session took approximately 15 minutes.

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Results

Mental arithmetic training progression. An AN-OVA with the session (1 to 6) as a within factor

was conducted on the mean solution times taken

by participants to solve a single-digit addition.

There was a significant effect of session, F(5,

125)�84.70, hp2�.77, pB.001. Indeed, there was

a clear training effect over sessions and partici-

pants were quicker to solve one-digit additions

during Session 6 (1945 ms) than during Session 1

(2940 ms), F(1, 25)�110.14, hp2�.82, pB.001.

An ANOVA with the same design as before

was performed on the mean solution times taken

by participants to solve a two-digit addition.

There was a significant effect of session, F(5,

125)�57.61,hp2�.70, pB.001. This effect of train-

ing was also reflected in the fact that the mean

solving time for a two-digit addition was shorter

during Session 6 (4192 ms) than during Session 1

(6525 ms), F(1, 25)�113.69, hp2�.82, pB.001.

Written arithmetic progression. A 2 (traininggroup: arithmetic and no-training)�2 (test phase:

pre-test vs. post-test) ANOVA with the first factor

as a between-subject measure and the last factor

as a repeated measure was performed on the

mean solution times taken by participants to solve

one addition from the French Kit (see Table 1).Mean solution times per written addition were

shorter during the post-test (9454 ms) than during

the pre-test (10513 ms), F(1, 50)�14.87, hp2�.23,

pB.001. There was no difference in the solution

times depending on the training group, FB1.

More importantly, the progression on the French

Kit did not differ as a function of the training

group, FB1. In fact, the two groups performed

better during the post-test than during the pre-

test: arithmetic, F(1, 50)�11.72, hp2�.19, pB

.001; and no-training, F(1, 50)�4.12, hp2�.08,

p�.04.

Discussion

First and in conformity with previous results inthe literature, we showed a clear practice effecton single-digit additions over sessions. Moreinterestingly, we showed as well that participantsimproved their performance on two-digit addi-tions through practice, even if those problemswere different from one day to the other. Thereare at least two different possible explanations forthis transfer. On the one hand, it is possible thattraining basic arithmetic facts (i.e., one-digitproblems) speeds up mental multidigit arithmeticsolving through increase of efficiency in arith-metic fact retrieval. Indeed, solving a complexmental addition requires the solver to reach aseries of intermediary results, some of which canbe directly retrieved from long-term memory. Onthe other hand, it could be that it is the practice ofgeneral mental arithmetic procedures that en-ables individuals to apply them to a broad rangeof problems that are not specifically trained.These general procedures could correspond toefficient rounding up, rounding down, or decom-position procedures (e.g., Lemaire & Callies,2009), to the storage of intermediate results,which is required both for simple and complexadditions (Thevenot, Barrouillet, & Fayol, 2001;Thevenot, Fanget, & Fayol, 2007), or to themonitoring of operation sequences (e.g., Semenzaet al., 1997). Of course, it could be the combina-tion of both factors that explains the possiblegeneralisation of practice to new two-digit addi-tions. On an educational point of view, thistransfer from a set of additions to other sets ofadditions shows that practice can be considered asan optimum method of learning. Of course, asalready suggested in the literature, long-termeffects of such training are expected if, in combi-nation to procedural training, conceptual knowl-edge is not neglected (Rittle-Johnson, Siegler, &Alibali, 2001).

Second, we showed that training mental arith-metic does not transfer to written arithmetic.More precisely, adults who received an arithmetictraining programme did not improve their perfor-mance more than adults who did not receive anytraining. As a consequence, the improvement inperformance in the arithmetic group can merelybe attributed to a test�retest advantage. There-fore, it seems that most of the procedures usedand reactivated by our participants during thetraining programme are irrelevant for written

TABLE 1

Mean solution times (in ms) per addition in the French

kit as a function of the phase of the experiment (pre- and

post-test) and the training group in Experiment 1

Pre-test Post-test

M SD M SD Progression

Arithmetic group 10,443 590 9114 485 1329

No-training group 10,582 590 9795 485 787

MENTAL AND WRITTEN ARITHMETIC 289

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arithmetic. A similar result was obtained by Imbo

and Vandierendonck (2008), who showed thatadults who were trained on simple addition did

not score better on the French Kit than adults

who did not receive any practice. However,

because their participants were tested on the

written arithmetic test only after practice and

not before, it was difficult to put forward definiteconclusions about transfer. Nevertheless, taken

together, these results comfort neuropsychologi-

cal research conclusions that written arithmetic is

a specific ability, which can be dissociated from

other numerical competencies. However, before atheoretical discussion can be engaged about the

relationship between written and mental calcula-

tion, it is necessary to examine the potential

transfer from written to mental arithmetic. It is

the object of the second experiment.

EXPERIMENT 2

Method

Participants

Fifty-eight students from the University of

Geneva took part in this experiment either forcourse credits or money compensation. In order

to test participants with similar arithmetic abil-

ities than in the previous experiment, we

recruited them from a database that specifies

their score on the French Kit. One half ofparticipants was assigned to the written arithmetic

training group and the other half to a control

group who did not receive any training. The

arithmetic group was composed of 29 students

with a mean score of 25.17 (SD�6.78;

range�11�42) and the no-training group wascomposed of 29 students with a mean score of

26.17 (SD�8.91; range�16�45) on the addition

subtest of the French Kit.

Material and procedure

The mental arithmetic test. Participants weretested in a pre- and post-tests on their mental

arithmetic abilities. The test consisted in two

blocks of one-digit addition problems and twoblocks of two-digit additions. The material and

procedure were strictly the same as for the first

training day in the previous experiment.

The written arithmetic training programme.Participants in the arithmetic group were trainedover a 6-day period and the last training dayalways took place before the post-test. In order toequalise the amount of written arithmetic practiceand the amount of mental arithmetic practice,participants had to solve four blocks of writtenmultidigit additions daily. They were all providedwith a timer and instructed to spend 4 minutes oneach block. Therefore, as in Experiment 1, eachtraining session took 16 minutes, without thebreaks that participants were allowed to takebetween blocks. As it can be seen in the resultsection, participants solved 144 problems onaverage across sessions. In order to match thetraining sessions as closely as possible to theFrench Kit (i.e., used as the pre- and post-testsin Experiment 1), each block consisted of addi-tions in columns involving three single- or two-digit numbers comprised between 1 and 99 (e.g.,44�9�51).

Results

Written arithmetic progression. An ANOVAwith the session (1 to 6) as a within factor wasconducted on the mean solution times taken byparticipants to solve one written addition. Therewas a significant effect of session, F(5,140)�40.50, hp

2�.60, pB.001. Indeed, therewas a clear training effect over sessions andparticipants were quicker to solve additions dur-ing Session 6 (6372 ms) than during Session 1(8587 ms), F(1, 28)�58.34, hp

2�.68, pB.001.

Mental arithmetic progression. A 2 (group:arithmetic training vs. no-training)�2 (phase:pre-test vs. post-test)�2 (type of addition: singlevs. two digit) ANOVA with the first factor as abetween-subject measure and the last factors asrepeated measures was performed on the meansolution times taken by participants to solve anaddition (see Figure 1).

Mean solution times were shorter during thepost-test (3033 ms) than during the pre-test (3540ms), F(1, 56)�159.46, hp

2�.74, pB.001, andshorter for simple than complex additions (1983ms and 4589 ms, respectively), F(1, 56)�402.89,hp

2�.88, pB.001. There was no difference in thesolution times depending on the group, FB1.However, and more importantly, the interaction

290 THEVENOT AND CASTEL

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between the training group and the testing phasewas significant, F(1, 56)�41.10, hp

2�.42, pB.001.In fact, both groups performed better after thanbefore the training programme, F(1, 56)�181.23,hp

2�.76, pB.001, and F(1, 56)�19.36, hp2�.26,

pB.001, for the arithmetic and the control group,respectively, but the progression was higher forthe arithmetic group (�765 ms) than for thecontrol group (�250 ms). Finally, these effectsinteracted with the type of addition, F(1,56)�16.69, hp

2�.23, pB.001, showing that thearithmetic training effect was more pronouncedfor two-digit addition than for single-digitaddition.

Discussion

The results of this experiment show that it ispossible to train written arithmetic through re-petitive practice over a 6-day period. Even if notsurprising, this result is noticeable because itshows that written arithmetic can still be im-proved in adults and that the lack of transfer frommental to written addition in the previous experi-ment was not due to ceiling effect. Furthermore,

and more interestingly, we showed here thatpractising written addition transfers to mentaladdition.

GENERAL DISCUSSION

The results of the two experiments reported inthis paper show that, whereas training mentalarithmetic does not transfer to written arithmetic,performance in mental arithmetic is improvedafter written arithmetic practice. Before conclu-sions can be drawn from these results, it isimportant to ensure that the mental and thewritten arithmetic tests were equally sensitive toimprovement. A strong argument for this assump-tion is that both written and mental tests weresensitive to test�retest effects. As a matter of fact,these effects were strictly the same for bothdomain of arithmetic. For the written test, parti-cipants from the control group took 10582 ms peraddition during the pre-test and 9795 ms peraddition during the post-test, which correspondsto a 7% progression. For simple mental addition,participants took 2148 ms per addition during thepre-test and 1997 ms during the post-test, which

1400

1600

1800

2000

2200

2400

Pre-test Post-test

Pre-test Post-test3400

3900

4400

4900

5400

5900

Solu

tion

times

(in

ms)

Solu

tion

times

(in

ms)

(a)

(b)

No training group

Training group

Figure 1. Mean solution times (and confidence intervals) per addition in the mental arithmetic test as a function of the phase of the

experiment (pre- and post-test), and the training group for simple addition (1a) and complex addition (1b) in Experiment 2.

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also corresponds to a 7% progression. Finally, forcomplex mental addition, participants took 4825ms per addition during the pre-test and 4477 msduring the post-test, which, again, also corre-sponds to a 7% progression. Because our differ-ent tests show the same sensitivity to the test�retest effect, they necessarily present the samesensitivity to practice.

We can therefore conclude that the unidirec-tional transfer that we observe implies that thetwo domains share common components that arethe main constituents of mental arithmetic butjust small constituents of written arithmetic. Asfar as the shared components are concerned andas already seen in our introduction, good perfor-mance in written and mental addition necessarilyrelies on quick and efficient fact retrieval oralgorithmic procedures. Therefore, our resultssuggest that fast retrieval or counting is the keyfor succeeding in mental calculation. However,because the transfer is not bidirectional, ourresults also suggest that such competencies arenot sufficient to succeed in written arithmetic.Indeed, besides fast calculations, written additionrequires individuals to apply the strategies taughtat school consisting in adding the units andthen the tens, using carrying when necessary(Figure 2). Written calculation would thereforedepend heavily on these specific taught proce-dures and adults’ performance would still bedriven by these formal solving routines. No suchstrict procedures are necessary to solve mentaladdition, which requires the solver to make use ofmore flexible strategies depending on problems’

characteristics (Lemaire & Callies, 2009). If anadult is asked to solve 59�47 mentally, he or shecan use the full decomposition or partial decom-position already described in our introduction,but can also figure out that 60�47 is 107 andtherefore that 59�47 is 107 �1. In fact, the beststrategies consist in decomposing the problem in aseries of subgoals that can be reached as effi-ciently as possible. As a matter of fact, efficientand adapted decompositions depend on thesolver’s arithmetic fact knowledge database. It isthen understandable that fact retrieval constitutesa central feature of mental addition.

Therefore, our results comfort neuropsycholo-gical studies that written calculation depends onspecific abilities that are dissociable from thoserequired in mental arithmetic (Semenza et al.,1997). However, these results could seem at oddswith the conclusions of Ginsburg, Posner, andRussell (1981), who showed that American ele-mentary school children often solve written addi-tion problems by mental regrouping procedures(e.g., 32�24 is 30�20�50�6�56). If, indeed,mental procedures can be applied to writtenarithmetic, then a bidirectional transfer betweenwritten and mental calculation should be ob-served. Nevertheless, a closer look at their resultsshows that children who were most likely to usemental strategies to solve written addition werethe youngest ones. More precisely, whereas8-year-old children applied mental strategies to21% of the written problems under study, 11-year-old children used mental strategies only for 4% ofthe same problems. This strongly suggests that

Figure 2. Shared and specific components in mental and written arithmetic.

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throughout schooling and practice, children be-come less flexible in their strategy use and

systematically engage in rigid routines.However, the strict appliance of routines does

not necessarily constitute a drawback in effi-ciency. Indeed, strategy adaptation would require

individuals to take the specificities of the written

arithmetic problems into account in order toquickly decide if the more economic way to solve

it would be the units and then tens routine or a

mental strategy. For example, 49�23 is probablysolved quicker if the solver figures out that it

corresponds to 50�23 �1 than if the solverengages in a strict routine consisting in adding

up 9 and 3, writing down 2, carrying the 1 into the

tens columns and finally writing down the 7 inorder to reach the answer 72. On the contrary,

23�26 is probably solved quicker when the

written routine is applied because no mentalstrategy can constitute efficient shortcut. How-

ever, selecting the most efficient strategy requires

the solver to pay attention to the characteristics ofthe problem before engaging in the solving

process and to constantly switch from a strategy

to the other, which is necessarily cognitivelydemanding (Luwel, Schillemans, Onghena, &

Verschaffel, 2008). Lemaire and Lecacheur

(2010) showed indeed that individuals obtainpoorer performance when they use different

strategies on consecutive trials than when they

use the same strategy and tend to repeat it oversuccessive problems. Therefore, further studies

will be necessary to determine if adults andchildren would benefit more from an adaptive

use of strategies in the domain of written arith-

metic, and consequently from better transfer frommental to written arithmetic, than from a strict

appliance of routines.

Original Manuscript received January 2011

Revised manuscript received July 2011

First published online November 2011

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